#### To determine

**To find:**

The limit as h→0 of the average value of f on the interval x, x+h.

#### Answer

fx

#### Explanation

**1) Concept:**

Use the mean value theorem for integrals

**2) Theorem:**

If f is continuous on [a, b], then there exists a number c in [a, b] such that

fc=fave =1b-a∫abfxdx In other words,

∫abfxdx=f(c)(b-a)

**3) Calculations:**

Since f is continuous,

By the mean value theorem for integrals,

fave =1b-a∫abfxdx

Limit as h→0 of the average value of f on interval x, x+h,

limh→0fave=limh→01x+h-x∫xx+hftdt

By the Fundamental Theorem of Calculus (2),

limh→0fave=limh→0Fx+h-Fxh, where Fx=∫axftdt.

But,limh→0Fx+h-Fxh=F'(x)

Therefore, by the Fundamental theorem of Calculus (1),

limh→0fave=F'x=fx

**Conclusion:**

The limit as h→0 of the average value of f on the interval x, x+h is fx.